If $A,B\subset H$ are closed subspaces and $\operatorname{codim}(A+B)<\infty$ do we have $A+B$ closed?

Question

Let $H$ be a Hilbert space and $A,B\subset H$ be two closed subspaces such that $\operatorname{codim}(A+B)<\infty $. I would be very surprised if it tuns out that $A+B$ is not necessarily closed. I've tried to use the closed graph theorem with the help of projections but without any success.